Category Archives: Number Theory

My first experience of problem setting. It feels good when you see people think about your problem and come up with better solutions than yours 😛
This problem is about math and primes. The link is this.

What is a Palindrome?
A palindrome is a word, which reads backwards the same as it does forwards. Well known examples are Anna or radar.
You can apply this principle to numbers. For instance 1001 or 69896 are palindromes.

Counting the Palindromes
All the digits are palindromes (1,2,3,…,9).

There are also 9 palindromes with two digits(11,22,33, …,99).

You can find to every two-digit number one, and only one number with three digits and with four digits.
For example: For the number 34 there are 343 and 3443.
You can conclude that there are 90 palindromes withthree and also 90 palindromes with four digits.

You can find to every three-digit number one, and only one number with five digits and with six digits.
For example: To the number 562 there are 56265 and 562265.
You can conclude that there are 900 palindromes withfive and 900 palindromes with six digits.

You have 9+9+90+90+900+900 = 1998 palindromes up to one million. That’s 0,1998 %. About every 500th number is a palindrome.

Position of the Palindromes
But they are not spread over all numbers regularly. This shows the picture below, which includes the first 1000 numbers.

I suppose that all products with the digit 1 are palindromes, if one.factor has at the most 9 digits.
All palindromes have the shape 123…..321.

Prime Numbers among the Palindromes
All palindromic primes with 3 digits:

101
131
151
181
191

313
353
373
383
.

727
757
787
797
.

919
929
.
.
.

There are no primes with 4 digits. They all have the factor 11. (Example:4554=4004+550=4×1001+550=4x91x11+11×50=11x(4×91+50)
There are 93 primes with 5 digits.
There are no primes with 6 digits. They all have the factor 11.
There are 668 primes with 7 digits.

196-Problem
Pick a number. Add the number, which you must read from the right to the left (mirror number), to the original number. Maybe the sum is a palindrome. If the sum isn’t a palindrome, add the mirror number of the sum to the sum. Maybe you have a palindrome now, otherwise repeat the process. Nearly all numbers have a palindrome in the end.
Example: 49 49+94=143 143+341= 484 !
There are some numbers, which have no palindromes. The lowest one is 196. But the proof is still missing.